Abstract
The third lesson was devoted to the decomposition of a signal propagating along a transmission line into orthogonal discrete modes. We introduced the basic concepts of wavelet theory. It is remarkable that a signal of finite energy can be decomposed onto a basis of continuous waves both localized in time and frequency. Each basic wave can be represented by a rectangle in the time-frequency plane, reminiscent of the staff of a musical score. The width of the rectangle corresponds to the time step of the base, and its height to the frequency step of the base. This discrete segmentation corresponds to a " first quantization " of the signals. Once these properties have been acquired, we can introduce the second quantization: this consists in declaring as conjugate variables the complex amplitudes of two discrete modes with the same position index in time, but with opposite indexes in frequency. This second quantization implies the existence of discrete states for the wave function describing the amplitude of the two modes. We thus arrive at the discrete states corresponding to photons, with the underlying wavelets constituting the " wave function " of photons. Note that this wave function cannot have all its moments finite in both frequency and time, like a Gaussian function, and at the same time serve as a pattern for a discrete basis.
